HANOI—The U.S.'s top-ranked military officer met his Vietnamese counterpart here Thursday, as the U.S. capitalizes on tensions between China and its Asia-Pacific neighbors to revamp its own regional alliances.

Gen. Martin Dempsey became the first chairman of the Joint Chiefs of Staff to visit the Southeast Asian country since the Vietnam War. Vietnam's Defense Ministry said in a statement that he and Gen. Do Ba Ty, the chief of...